An absorbance of 1 of proteins at 280 nm (absorbance of tryptophane and tyrosin) equals around 1 mg/ml. This is true for a heterogenic mixture of proteins, of course this will differ between different proteins. I just tested this with an enzyme (glutathione reductase in yeast) and if I did the calculation correct it differed less than 1 % of this value.

I was hoping someone knows more about this.

What does this value depend on? Protein size?

And the fact that A 1 ≈ 1 mg/ml, is that just a coincidence, or is is somehow man-made through equations etc?

How much the protein absorbs light at 280nm will be a function of how much tryptophan and tyrosine are in the protein. A protein composed entirely of polyarginine would not absorb at 280nm. From this you can see that the coincidence of measuring an absorbance of 1 at 1 mg/ml protein is just that, a coincidence. A protein mostly composed of tryptophan and tyrosine would have an absorbance of 1 at a far lower concentration than 1 mg/ml. Proteins tend toward average values for the distribution of amino acids so a rule of thumb like A=1 for 1 mg/ml might be a reasonable first approximation but, for an accurate measurement, you would need to know the molar extinction coefficient for the particular protein of interest at 260 nm.

The absorbance of 280 nm light occurs because electrons in the conjugated pi orbitals of the aromatic amino acids can undergo energy-level transitions with energy differences equaling the energy in a quantum of 280 nm light. The 280 nm absorbance depends on the presence of those conjugated pi orbitals, that is, on the presence of tryptophan and tyrosine.